Double Pendulum 

In[7]:=

X Animate[Graphics[{{PointSize[.025], {Red, Point[{x1[t], y1[t]}]}, {Blue, Point[{x2[t], y2[t]}]}, Line[{{0, 0}, {x1[t], y1[t]}, {x2[t], y2[t]}}]} /. soldp, {Gray, Line[Map[Function[Evaluate[{x2[#], y2[#]} /. soldp]], Range[0, t, 0.025]]]}}, PlotRange -> {{-2, 2}, {-2, 0}}, Axes -> True, Ticks -> False, ImageSize -> 500], {t, 0, 10, .025}, SaveDefinitions -> True]

In[8]:=

X With[{t = 3.7}, Graphics[{{Thick, Line[{{0, 0}, {x1[t], y1[t]}, {x2[t], y2[t]}}], PointSize[.025], {Red, Point[{x1[t], y1[t]}]}, {Blue, Point[{x2[t], y2[t]}]},} /. soldp, {Gray, Line[Map[Function[Evaluate[{x2[#], y2[#]} /. soldp]], Range[0, t, 0.025]]]}}, PlotRange -> {{-1.6, 1.6}, {-2, 0}}, Axes -> True, Ticks -> False, ImageSize -> 500]]

Play Animation » Stop Animation »

Model the motion of a double pendulum in Cartesian coordinates.

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Derive the governing equations using Newton's second law of motion, and .

In[1]:=

X deqns = {Subscript[m, 1] x1''[t] == (\[Lambda]1[t]/Subscript[l, 1]) x1[ t] - (\[Lambda]2[t]/Subscript[l, 2]) (x2[t] - x1[t]), Subscript[m, 1] y1''[t] == (\[Lambda]1[t]/Subscript[l, 1]) y1[ t] - (\[Lambda]2[t]/Subscript[l, 2]) (y2[t] - y1[t]) - Subscript[m, 1] g, Subscript[m, 2] x2''[t] == (\[Lambda]2[t]/Subscript[l, 2]) (x2[t] - x1[t]), Subscript[m, 2] y2''[t] == (\[Lambda]2[t]/Subscript[l, 2]) (y2[t] - y1[t]) - Subscript[m, 2] g};

The lengths of the pendulum rods are fixed. These are expressed as algebraic constraints.

In[2]:=

X aeqns = {x1[t]^2 + y1[t]^2 == Subscript[l, 1]^2, (x2[t] - x1[t])^2 + (y2[t] - y1[t])^2 == Subscript[l, 2]^2};

Specify the initial state of the system as initial conditions.

In[3]:=

X ics = {x1[0] == 1, y1[0] == 0, x1'[0] == 0, y1'[0] == 0, x2[0] == 1, y2[0] == -1, x2'[0] == 0, y2'[0] == 0};

Specify the physical parameters for the pendulum system.

In[4]:=

X params = {g -> 9.81, Subscript[m, 1] -> 1, Subscript[m, 2] -> 1, Subscript[l, 1] -> 1, Subscript[l, 2] -> 1};

Solve and visualize the system.

In[5]:=

X soldp = First[ NDSolve[{deqns, aeqns, ics} /. params, {x1, y1, x2, y2, \[Lambda]1, \[Lambda]2}, {t, 0, 15}, Method -> {"IndexReduction" -> {Automatic, "IndexGoal" -> 0}}]];

In[6]:=

X ParametricPlot[ Evaluate[{{x1[t], y1[t]}, {x2[t], y2[t]}} /. soldp], {t, 0, 15}, PlotStyle -> {Red, Blue}, ImageSize -> Medium, PlotLegends -> {"Trajectory of pendulum 1", "Trajectory of pendulum 2"}]

Out[6]=